Cartesian product and symmetric difference

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Discussion Overview

The discussion revolves around proving the equality of the Cartesian product of a set with the symmetric difference of two other sets. Participants explore the relationship between the sets A, B, and C, specifically focusing on the expression Ax(BΔC) and its equivalence to (AxB) Δ (AxC).

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Technical explanation

Main Points Raised

  • One participant begins by defining an arbitrary element of Ax(BΔC) and expresses uncertainty about how to proceed with the proof.
  • Another participant confirms the initial steps taken and questions the conditions under which an element belongs to (AxB) Δ (AxC).
  • A later reply suggests that the participant may have a clearer understanding and outlines a reasoning path to show that Ax(BΔC) is a subset of (AxB) Δ (AxC).
  • Further, the discussion includes a suggestion to work backwards from the established subset to complete the proof.

Areas of Agreement / Disagreement

Participants generally agree on the correctness of the initial steps in the proof, but the discussion remains unresolved regarding the complete proof and the conditions necessary for the equivalence to hold.

Contextual Notes

Participants express uncertainty about the next steps in the proof and the conditions for the subset relationships, indicating potential limitations in their current understanding.

fatineouahbi
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Let A,B,C be three sets . Prove Ax(BΔC)= (AxB) Δ (AxC)

I tried to start with this :

Let p be an arbitrary element of Ax(BΔC)
then p=(x,y) such that x ∈ A and y ∈ (BΔC)
x ∈ A and (y∈ B\C or y∈ C\B)
(x ∈ A and y ∈ B\C) or (x ∈ A and y ∈ C\B)

But I don't know how to continue or if I should even start with this .
 
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fatineouahbi said:
Let A,B,C be three sets . Prove Ax(BΔC)= (AxB) Δ (AxC)

I tried to start with this :

Let p be an arbitrary element of Ax(BΔC)
then p=(x,y) such that x ∈ A and y ∈ (BΔC)
x ∈ A and (y∈ B\C or y∈ C\B)
(x ∈ A and y ∈ B\C) or (x ∈ A and y ∈ C\B)

But I don't know how to continue or if I should even start with this .

It is right so far.When does it hold that $p \in (A \times B) \triangle (A \times C)$ ?
 
evinda said:
It is right so far.When does it hold that $p \in (A \times B) \triangle (A \times C)$ ?

Hello :) Thank you , I think I may get it now ?

(x ∈ A and y ∈ B\C) or (x ∈ A and y ∈ C\B)
then p ∈ Ax(B\C) or p ∈ Ax(C\B)
then p ∈ (AxB) \ (AxC) or p ∈ (AxC) \ (AxB)
thus p ∈ (AxB) △ (AxC)
then Ax(BΔC) ‎⊂ (AxB) Δ (AxC)

Then I'll just try to go backwards maybe ?
 
fatineouahbi said:
Hello :) Thank you , I think I may get it now ?

(x ∈ A and y ∈ B\C) or (x ∈ A and y ∈ C\B)
then p ∈ Ax(B\C) or p ∈ Ax(C\B)
then p ∈ (AxB) \ (AxC) or p ∈ (AxC) \ (AxB)
thus p ∈ (AxB) △ (AxC)
then Ax(BΔC) ‎⊂ (AxB) Δ (AxC)
Well done, you are right :)

fatineouahbi said:
Then I'll just try to go backwards maybe ?
Yes, you pick an element in $(A \times B)\triangle (A \times C)$ and you need to show that it is also in $A \times (B \triangle C)$.
 
evinda said:
Well done, you are right :)

Yes, you pick an element in $(A \times B)\triangle (A \times C)$ and you need to show that it is also in $A \times (B \triangle C)$.

Thank you so much !
 

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